Monogamous subvarieties of the nilpotent cone

Abstract

Let $G$ be a reductive algebraic group over an algebraically closed field $k$ of prime characteristic not $2$, whose Lie algebra is denoted $\mathfrak{g}$. We call a subvariety $\mathfrak{X}$ of the nilpotent cone $N\subset \mathfrak{g}$ monogamous if for every $e\in \mathfrak{X}$, the ${\mathfrak{sl}}_2$-triples $(e,h,f)$ with $f\in \mathfrak{X}$ are conjugate under the centraliser $C_G(e)$. Building on work by the first two authors, we show there is a unique maximal closed $G$-stable monogamous subvariety $V\subset N$ and that it is an orbit closure, hence irreducible. We show that $V$ can also be characterised in terms of Serre's $G$-complete reducibility.

16 pages

References in corpus (4)

• On the minimal modules for exceptional Lie algebras: Jordan blocks and stabilisers
• Completely reducible Lie subalgebras
• The Jacobson--Morozov theorem and complete reducibility of Lie subalgebras
• Normality of Orthogonal and Sympletic Nilpotent Orbit Closures in Positive Characteristic